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Properties

Label 20.0.69654621282...6096.6
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $24.67$
Ramified primes $2, 11, 23$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T751

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3433, -7824, 13150, -11924, 12027, -8424, 8288, -4347, 2621, -913, 590, 77, -83, 23, 9, 2, 12, -11, 3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 3*x^18 - 11*x^17 + 12*x^16 + 2*x^15 + 9*x^14 + 23*x^13 - 83*x^12 + 77*x^11 + 590*x^10 - 913*x^9 + 2621*x^8 - 4347*x^7 + 8288*x^6 - 8424*x^5 + 12027*x^4 - 11924*x^3 + 13150*x^2 - 7824*x + 3433)
 
gp: K = bnfinit(x^20 - x^19 + 3*x^18 - 11*x^17 + 12*x^16 + 2*x^15 + 9*x^14 + 23*x^13 - 83*x^12 + 77*x^11 + 590*x^10 - 913*x^9 + 2621*x^8 - 4347*x^7 + 8288*x^6 - 8424*x^5 + 12027*x^4 - 11924*x^3 + 13150*x^2 - 7824*x + 3433, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 3 x^{18} - 11 x^{17} + 12 x^{16} + 2 x^{15} + 9 x^{14} + 23 x^{13} - 83 x^{12} + 77 x^{11} + 590 x^{10} - 913 x^{9} + 2621 x^{8} - 4347 x^{7} + 8288 x^{6} - 8424 x^{5} + 12027 x^{4} - 11924 x^{3} + 13150 x^{2} - 7824 x + 3433 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6965462128295683654948996096=2^{10}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.67$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{61912693439121229667624335605859404707} a^{19} + \frac{834126907724845522341299365799110719}{61912693439121229667624335605859404707} a^{18} + \frac{24034867747001776290827034020094486514}{61912693439121229667624335605859404707} a^{17} - \frac{23065269262180474610543448853690542926}{61912693439121229667624335605859404707} a^{16} + \frac{24646834484229129500198086050764982345}{61912693439121229667624335605859404707} a^{15} + \frac{3824346591671262280665727473166939162}{61912693439121229667624335605859404707} a^{14} - \frac{14848467646966760073551983379952791495}{61912693439121229667624335605859404707} a^{13} - \frac{10182324296846314734873334583443984038}{61912693439121229667624335605859404707} a^{12} - \frac{18591775116872732421010330699204469543}{61912693439121229667624335605859404707} a^{11} - \frac{2628973172240297858190444231033273531}{61912693439121229667624335605859404707} a^{10} - \frac{3425246247263081323960779437728197530}{61912693439121229667624335605859404707} a^{9} - \frac{367121559744509586970728856880267341}{61912693439121229667624335605859404707} a^{8} + \frac{14213926743951303280635484550883000611}{61912693439121229667624335605859404707} a^{7} - \frac{29167194670696939998219956113518334251}{61912693439121229667624335605859404707} a^{6} + \frac{14682383955209997499454025855754695586}{61912693439121229667624335605859404707} a^{5} + \frac{842693658471067649917242989904076075}{2691856236483531724679318939385191509} a^{4} + \frac{16978676092625835674664088574977316571}{61912693439121229667624335605859404707} a^{3} - \frac{6654877804492504351052923320089008191}{61912693439121229667624335605859404707} a^{2} - \frac{9945237900007239561026322913007825141}{61912693439121229667624335605859404707} a - \frac{21665080158055762150913494960272583957}{61912693439121229667624335605859404707}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 158254.649471 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T751:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n751 are not computed
Character table for t20n751 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.4.4930254263.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.10.10.5$x^{10} - 9 x^{8} + 50 x^{6} - 50 x^{4} + 45 x^{2} - 5$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$